Tractability of Function Approximation with Product Kernels

Abstract

This article studies the problem of approximating functions belonging to a Hilbert space ${\mathscr {H}}_d$ with a reproducing kernel of the form

$${\widetilde{K}}_d({\varvec{x}},{\varvec{t}}):=\prod _{\ell =1}^d \left( 1-\alpha _\ell ^2+\alpha _\ell ^2K_{\gamma _\ell }(x_\ell ,t_\ell )\right) \ \ \ \text{ for } \text{ all } \ \ \ {\varvec{x}},{\varvec{t}}\in {\mathbb {R}}^d.$$

The $\alpha _\ell \in [0,1]$ are scale parameters, and the $\gamma _\ell >0$ are sometimes called shape parameters. The reproducing kernel $K_{\gamma }$ corresponds to some Hilbert space of functions defined on ${\mathbb {R}}$. The kernel ${\widetilde{K}}_d$ generalizes the anisotropic Gaussian reproducing kernel, whose tractability properties have been established in the literature. We present sufficient conditions on $\{\alpha _\ell \gamma _\ell \}_{\ell =1}^{\infty }$ under which function approximation problems on ${\mathscr {H}}_d$ are polynomially tractable. The exponent of strong polynomial tractability arises from bounds on the eigenvalues of positive definite linear operators.